Tag Archives: amplitudes

The Conference Dilemma: Freshness vs. Breadth

Back in 2017, I noticed something that should have struck me as a little odd. My sub-field has a big yearly conference, called Amplitudes, that brings in everyone who works on our kind of research. Amplitudes 2017 was fun, but not “fresh”: most people talked about work they had already published. A smaller conference I went to that year, called QCD Meets Gravity, was much “fresher”: a lot of discussion of work in progress and work “hot off the presses”.

At the time, I chalked the difference up to timing: it was a few months later, and people happened to have projects that matured around then. But I realized recently there’s another reason, one why you would expect bigger conferences to have less fresh content.

See, I’ve recently been on the other “side of the curtain”: I was an organizer for Amplitudes last year. And I noticed one big obstacle to having fresh content: the timeframe.

The bigger a conference is, the longer in advance you need to invite speakers. It’s a bigger task to organize everyone, to make sure travel and hotels and raw availability works, that everyone has time to prepare their talks and you have a nice full (but not too full) schedule. So when we started asking people, we didn’t know what the “freshest” work was going to be. We had recommendations from our scientific committee (a group of experts in the subfield whose job is to suggest speakers), but in practice the goal is more one of breadth than freshness: we needed to make sure that everybody in our community was represented.

A smaller conference can get around this. It can be organized a bit later, so the organizers have more information about new developments. It covers a smaller area, so the organizers have more information about new hot topics and unpublished results. And it typically invites most of the sub-community anyway, so you’re guaranteed to cover the hot new stuff just by raw completeness.

This doesn’t mean small conferences are “just better” or anything like that. Breadth is genuinely useful: a big conference covering a whole subfield is great for bringing a community together, getting everyone on a shared page and expanding their horizons. There’s a real tradeoff between those goals and getting a conference with the latest progress. It’s not a fixed tradeoff, we can improve both goals at once (I think at Amplitudes we as organizers could have been better at highlighting unpublished work), but we still have to make choices of what to emphasize.

Carving Out the Possible

If you imagine a particle physicist, you probably picture someone spending their whole day dreaming up new particles. They figure out how to test those particles in some big particle collider, and for a lucky few their particle gets discovered and they get a Nobel prize.

Occasionally, a wiseguy asks if we can’t just cut out the middleman. Instead of dreaming up particles to test, why don’t we just write down every possible particle and test for all of them? It would save the Nobel committee a lot of money at least!

It turns out, you can sort of do this, through something called Effective Field Theory. An Effective Field Theory is a type of particle physics theory that isn’t quite true: instead, it’s “effectively” true, meaning true as long as you don’t push it too far. If you test it at low energies and don’t “zoom in” too much then it’s fine. Crank up your collider energy high enough, though, and you expect the theory to “break down”, revealing new particles. An Effective Field Theory lets you “hide” unknown particles inside new interactions between the particles we already know.

To help you picture how this works, imagine that the pink and blue lines here represent familiar particles like electrons and quarks, while the dotted line is a new particle somebody dreamed up. (The picture is called a Feynman diagram, if you don’t know what that is check out this post.)

In an Effective Field Theory, we “zoom out”, until the diagram looks like this:

Now we’ve “hidden” the new particle. Instead, we have a new type of interaction between the particles we already know.

So instead of writing down every possible new particle we can imagine, we only have to write down every possible interaction between the particles we already know.

That’s not as hard as it sounds. In part, that’s because not every interaction actually makes sense. Some of the things you could write down break some important rules. They might screw up cause and effect, letting something happen before its cause instead of after. They might screw up probability, giving you a formula for the chance something happens that gives a number greater than 100%.

Using these rules you can play a kind of game. You start out with a space representing all of the interactions you can imagine. You begin chipping at it, carving away parts that don’t obey the rules, and you see what shape is left over. You end up with plots that look a bit like carving a ham.

People in my subfield are getting good at this kind of game. It isn’t quite our standard fare: usually, we come up with tricks to make calculations with specific theories easier. Instead, many groups are starting to look at these general, effective theories. We’ve made friends with groups in related fields, building new collaborations. There still isn’t one clear best way to do this carving, so each group manages to find a way to chip a little farther. Out of the block of every theory we could imagine, we’re carving out a space of theories that make sense, theories that could conceivably be right. Theories that are worth testing.

Of Snowmass and SAGEX

arXiv-watchers might have noticed an avalanche of papers with the word Snowmass in the title. (I contributed to one of them.)

Snowmass is a place, an area in Colorado known for its skiing. It’s also an event in that place, the Snowmass Community Planning Exercise for the American Physical Society’s Division of Particles and Fields. In plain terms, it’s what happens when particle physicists from across the US get together in a ski resort to plan their future.

Usually someone like me wouldn’t be involved in that. (And not because it’s a ski resort.) In the past, these meetings focused on plans for new colliders and detectors. They got contributions from experimentalists, and a few theorists heavily focused on their work, but not the more “formal” theorists beyond.

This Snowmass is different. It’s different because of Corona, which changed it from a big meeting in a resort to a spread-out series of meetings and online activities. It’s also different because they invited theorists to contribute, and not just those interested in particle colliders. The theorists involved study everything from black holes and quantum gravity to supersymmetry and the mathematics of quantum field theory. Groups focused on each topic submit “white papers” summarizing the state of their area. These white papers in turn get organized and summarized into a few subfields, which in turn contribute to the planning exercise. No-one I’ve talked to is entirely clear on how this works, how much the white papers will actually be taken into account or by whom. But it seems like a good chance to influence US funding agencies, like the Department of Energy, and see if we can get them to prioritize our type of research.

Europe has something similar to Snowmass, called the European Strategy for Particle Physics. It also has smaller-scale groups, with their own purposes, goals, and funding sources. One such group is called SAGEX: Scattering Amplitudes: from Geometry to EXperiment. SAGEX is an Innovative Training Network, an organization funded by the EU to train young researchers, in this case in scattering amplitudes. Its fifteen students are finishing their PhDs and ready to take the field by storm. Along the way, they spent a little time in industry internships (mostly at Maple and Mathematica), and quite a bit of time working on outreach.

They have now summed up that outreach work in an online exhibition. I’ve had fun exploring it over the last couple days. They’ve got a lot of good content there, from basic explanations of relativity and quantum mechanics, to detailed games involving Feynman diagrams and associahedra, to a section that uses solitons as a gentle introduction to integrability. If you’re in the target audience, you should check it out!

Geometry and Geometry

Last week, I gave the opening lectures for a course on scattering amplitudes, the things we compute to find probabilities in particle physics. After the first class, one of the students asked me if two different descriptions of these amplitudes, one called CHY and the other called the amplituhedron, were related. There does happen to be a connection, but it’s a bit subtle and indirect, not the sort of thing the student would have been thinking of. Why then, did he think they might be related? Well, he explained, both descriptions are geometric.

If you’ve been following this blog for a while, you’ve seen me talk about misunderstandings. There are a lot of subtle ways a smart student can misunderstand something, ways that can be hard for a teacher to recognize. The right question, or the right explanation, can reveal what’s going on. Here, I think the problem was that there are multiple meanings of geometry.

One of the descriptions the student asked about, CHY, is related to string theory. It describes scattering particles in terms of the path of a length of string through space and time. That path draws out a surface called a world-sheet, showing all the places the string touches on its journey. And that picture, of a wiggly surface drawn in space and time, looks like what most people think of as geometry: a “shape” in a pretty normal sense, which here describes the physics of scattering particles.

The other description, the amplituhedron, also uses geometric objects to describe scattering particles. But the “geometric objects” here are much more abstract. A few of them are familiar: straight lines, the area between them forming shapes on a plane. Most of them, though are generalizations of this: instead of lines on a plane, they have higher dimensional planes in higher dimensional spaces. These too get described as geometry, even though they aren’t the “everyday” geometry you might be familiar with. Instead, they’re a “natural generalization”, something that, once you know the math, is close enough to that “everyday” geometry that it deserves the same name.

This week, two papers presented a totally different kind of geometric description of particle physics. In those papers, “geometric” has to do with differential geometry, the mathematics behind Einstein’s theory of general relativity. The descriptions are geometric because they use the same kinds of building-blocks of that theory, a metric that bends space and time. Once again, this kind of geometry is a natural generalization of the everyday notion, but now in once again a different way.

All of these notions of geometry do have some things in common, of course. Maybe you could even write down a definition of “geometry” that includes all of them. But they’re different enough that if I tell you that two descriptions are “geometric”, it doesn’t tell you all that much. It definitely doesn’t tell you the two descriptions are related.

It’s a reasonable misunderstanding, though. It comes from a place where, used to “everyday” geometry, you expect two “geometric descriptions” of something to be similar: shapes moving in everyday space, things you can directly compare. Instead, a geometric description can be many sorts of shape, in many sorts of spaces, emphasizing many sorts of properties. “Geometry” is just a really broad term.

Classicality Has Consequences

Last week, I mentioned some interesting new results in my corner of physics. I’ve now finally read the two papers and watched the recorded talk, so I can satisfy my frustrated commenters.

Quantum mechanics is a very cool topic and I am much less qualified than you would expect to talk about it. I use quantum field theory, which is based on quantum mechanics, so in some sense I use quantum mechanics every day. However, most of the “cool” implications of quantum mechanics don’t come up in my work. All the debates about whether measurement “collapses the wavefunction” are irrelevant when the particles you measure get absorbed in a particle detector, never to be seen again. And while there are deep questions about how a classical world emerges from quantum probabilities, they don’t matter so much when all you do is calculate those probabilities.

They’ve started to matter, though. That’s because quantum field theorists like me have recently started working on a very different kind of problem: trying to predict the output of gravitational wave telescopes like LIGO. It turns out you can do almost the same kind of calculation we’re used to: pretend two black holes or neutron stars are sub-atomic particles, and see what happens when they collide. This trick has grown into a sub-field in its own right, one I’ve dabbled in a bit myself. And it’s gotten my kind of physicists to pay more attention to the boundary between classical and quantum physics.

The thing is, the waves that LIGO sees really are classical. Any quantum gravity effects there are tiny, undetectably tiny. And while this doesn’t have the implications an expert might expect (we still need loop diagrams), it does mean that we need to take our calculations to a classical limit.

Figuring out how to do this has been surprisingly delicate, and full of unexpected insight. A recent example involves two papers, one by Andrea Cristofoli, Riccardo Gonzo, Nathan Moynihan, Donal O’Connell, Alasdair Ross, Matteo Sergola, and Chris White, and one by Ruth Britto, Riccardo Gonzo, and Guy Jehu. At first I thought these were two groups happening on the same idea, but then I noticed Riccardo Gonzo on both lists, and realized the papers were covering different aspects of a shared story. There is another group who happened upon the same story: Paolo Di Vecchia, Carlo Heissenberg, Rodolfo Russo and Gabriele Veneziano. They haven’t published yet, so I’m basing this on the Gonzo et al papers.

The key question each group asked was, what does it take for gravitational waves to be classical? One way to ask the question is to pick something you can observe, like the strength of the field, and calculate its uncertainty. Classical physics is deterministic: if you know the initial conditions exactly, you know the final conditions exactly. Quantum physics is not. What should happen is that if you calculate a quantum uncertainty and then take the classical limit, that uncertainty should vanish: the observation should become certain.

Another way to ask is to think about the wave as made up of gravitons, particles of gravity. Then you can ask how many gravitons are in the wave, and how they are distributed. It turns out that you expect them to be in a coherent state, like a laser, one with a very specific distribution called a Poisson distribution: a distribution in some sense right at the border between classical and quantum physics.

The results of both types of questions were as expected: the gravitational waves are indeed classical. To make this work, though, the quantum field theory calculation needs to have some surprising properties.

If two black holes collide and emit a gravitational wave, you could depict it like this:

All pictures from arXiv:2112.07556

where the straight lines are black holes, and the squiggly line is a graviton. But since gravitational waves are made up of multiple gravitons, you might ask, why not depict it with two gravitons, like this?

It turns out that diagrams like that are a problem: they mean your two gravitons are correlated, which is not allowed in a Poisson distribution. In the uncertainty picture, they also would give you non-zero uncertainty. Somehow, in the classical limit, diagrams like that need to go away.

And at first, it didn’t look like they do. You can try to count how many powers of Planck’s constant show up in each diagram. The authors do that, and it certainly doesn’t look like it goes away:

An example from the paper with Planck’s constants sprinkled around

Luckily, these quantum field theory calculations have a knack for surprising us. Calculate each individual diagram, and things look hopeless. But add them all together, and they miraculously cancel. In the classical limit, everything combines to give a classical result.

You can do this same trick for diagrams with more graviton particles, as many as you like, and each time it ought to keep working. You get an infinite set of relationships between different diagrams, relationships that have to hold to get sensible classical physics. From thinking about how the quantum and classical are related, you’ve learned something about calculations in quantum field theory.

That’s why these papers caught my eye. A chunk of my sub-field is needing to learn more and more about the relationship between quantum and classical physics, and it may have implications for the rest of us too. In the future, I might get a bit more qualified to talk about some of the very cool implications of quantum mechanics.

Science, Gifts Enough for Lifetimes

Merry Newtonmas, Everyone!

In past years, I’ve compared science to a gift: the ideal gift for the puzzle-fan, one that keeps giving new puzzles. I think people might not appreciate the scale of that gift, though.

Bigger than all the creative commons Wikipedia images

Maybe you’ve heard the old joke that studying for a PhD means learning more and more about less and less until you know absolutely everything about nothing at all. This joke is overstating things: even when you’ve specialized down to nothing at all, you still won’t know everything.

If you read the history of science, it might feel like there are only a few important things going on at a time. You notice the simultaneous discoveries, like calculus from Newton and Liebniz and natural selection from Darwin and Wallace. You can get the impression that everyone was working on a few things, the things that would make it into the textbooks. In fact, though, there was always a lot to research, always many interesting things going on at once. As a scientist, you can’t escape this. Even if you focus on your own little area, on a few topics you care about, even in a small field, there will always be more going on than you can keep up with.

This is especially clear around the holiday season. As everyone tries to get results out before leaving on vacation, there is a tidal wave of new content. I have five papers open on my laptop right now (after closing four or so), and some recorded talks I keep meaning to watch. Two of the papers are the kind of simultaneous discovery I mentioned: two different groups noticing that what might seem like an obvious fact – that in classical physics, unlike in quantum, one can have zero uncertainty – has unexpected implications for our kind of calculations. (A third group got there too, but hasn’t published yet.) It’s a link I would never have expected, and with three groups coming at it independently you’d think it would be the only thing to pay attention to: but even in the same sub-sub-sub-field, there are other things going on that are just as cool! It’s wild, and it’s not some special quirk of my area: that’s science, for all us scientists. No matter how much you expect it to give you, you’ll get more, lifetimes and lifetimes worth. That’s a Newtonmas gift to satisfy anyone.

Calculations of the Past

Last week was a birthday conference for one of the pioneers of my sub-field, Ettore Remiddi. I wasn’t there, but someone who was pointed me to some of the slides, including a talk by Stefano Laporta. For those of you who didn’t see my post a few weeks back, Laporta was one of Remiddi’s students, who developed one of the most important methods in our field and then vanished, spending ten years on an amazingly detailed calculation. Laporta’s talk covers more of the story, about what it was like to do precision calculations in that era.

“That era”, the 90’s through 2000’s, witnessed an enormous speedup in computers, and a corresponding speedup in what was possible. Laporta worked with Remiddi on the three-loop electron anomalous magnetic moment, something Remiddi had been working on since 1969. When Laporta joined in 1989, twenty-one of the seventy-two diagrams needed had still not been computed. They would polish them off over the next seven years, before Laporta dove in to four loops. Twenty years later, he had that four-loop result to over a thousand digits.

One fascinating part of the talk is seeing how the computational techniques change over time, as language replaces language and computer clusters get involved. As a student, Laporta learns a lesson we all often need: that to avoid mistakes, we need to do as little by hand as possible, even for something as simple as copying a one-line formula. Looking at his review of others’ calculations, it’s remarkable how many theoretical results had to be dramatically corrected a few years down the line, and how much still might depend on theoretical precision.

Another theme was one of Remiddi suggesting something and Laporta doing something entirely different, and often much more productive. Whether it was using the arithmetic-geometric mean for an elliptic integral instead of Gaussian quadrature, or coming up with his namesake method, Laporta spent a lot of time going his own way, and Remiddi quickly learned to trust him.

There’s a lot more in the slides that’s worth reading, including a mention of one of this year’s Physics Nobelists. The whole thing is an interesting look at what it takes to press precision to the utmost, and dedicate years to getting something right.

In Uppsala for Elliptics 2021

I’m in Uppsala in Sweden this week, at an actual in-person conference.

With actual blackboards!

Elliptics started out as a series of small meetings of physicists trying to understand how to make sense of elliptic integrals in calculations of colliding particles. It grew into a full-fledged yearly conference series. I organized last year, which naturally was an online conference. This year though, the stage was set for Uppsala University to host in person.

I should say mostly in person. It’s a hybrid conference, with some speakers and attendees joining on Zoom. Some couldn’t make it because of travel restrictions, or just wanted to be cautious about COVID. But seemingly just as many had other reasons, like teaching schedules or just long distances, that kept them from coming in person. We’re all wondering if this will become a long-term trend, where the flexibility of hybrid conferences lets people attend no matter their constraints.

The hybrid format worked better than expected, but there were still a few kinks. The audio was particularly tricky, it seemed like each day the organizers needed a new microphone setup to take questions. It’s always a little harder to understand someone on Zoom, especially when you’re sitting in an auditorium rather than focused on your own screen. Still, technological experience should make this work better in future.

Content-wise, the conference began with a “mini-school” of pedagogical talks on particle physics, string theory, and mathematics. I found the mathematical talks by Erik Panzer particularly nice, it’s a topic I still feel quite weak on and he laid everything out in a very clear way. It seemed like a nice touch to include a “school” element in the conference, though I worry it ate too much into the time.

The rest of the content skewed more mathematical, and more string-theoretic, than these conferences have in the past. The mathematical content ranged from intriguing (including an interesting window into what it takes to get high-quality numerics) to intimidatingly obscure (large commutative diagrams, category theory on the first slide). String theory was arguably under-covered in prior years, but it felt over-covered this year. With the particle physics talks focusing on either general properties with perhaps some connections to elliptics, or to N=4 super Yang-Mills, it felt like we were missing the more “practical” talks from past conferences, where someone was computing something concrete in QCD and told us what they needed. Next year is in Mainz, so maybe those talks will reappear.

Stop Listing the Amplituhedron as a Competitor of String Theory

The Economist recently had an article (paywalled) that meandered through various developments in high-energy physics. It started out talking about the failure of the LHC to find SUSY, argued this looked bad for string theory (which…not really?) and used it as a jumping-off point to talk about various non-string “theories of everything”. Peter Woit quoted it a few posts back as kind of a bellwether for public opinion on supersymmetry and string theory.

The article was a muddle, but a fairly conventional muddle, explaining or mis-explaining things in roughly the same way as other popular physics pieces. For the most part that didn’t bug me, but one piece of the muddle hit a bit close to home:

The names of many of these [non-string theories of everything] do, it must be conceded, torture the English language. They include “causal dynamical triangulation”, “asymptotically safe gravity”, “loop quantum gravity” and the “amplituhedron formulation of quantum theory”.

I’ve posted about the amplituhedron more than a few times here on this blog. Out of every achievement of my sub-field, it has most captured the public imagination. It’s legitimately impressive, a way to translate calculations of probabilities of collisions of fundamental particles (in a toy model, to be clear) into geometrical objects. What it isn’t, and doesn’t pretend to be, is a theory of everything.

To be fair, the Economist piece admits this:

Most attempts at a theory of everything try to fit gravity, which Einstein describes geometrically, into quantum theory, which does not rely on geometry in this way. The amplituhedron approach does the opposite, by suggesting that quantum theory is actually deeply geometric after all. Better yet, the amplituhedron is not founded on notions of spacetime, or even statistical mechanics. Instead, these ideas emerge naturally from it. So, while the amplituhedron approach does not as yet offer a full theory of quantum gravity, it has opened up an intriguing path that may lead to one.

The reasoning they have leading up to it has a few misunderstandings anyway. The amplituhedron is geometrical, but in a completely different way from how Einstein’s theory of gravity is geometrical: Einstein’s gravity is a theory of space and time, the amplituhedron’s magic is that it hides space and time behind a seemingly more fundamental mathematics.

This is not to say that the amplituhedron won’t lead to insights about gravity. That’s a big part of what it’s for, in the long-term. Because the amplituhedron hides the role of space and time, it might show the way to theories that lack them altogether, theories where space and time are just an approximation for a more fundamental reality. That’s a real possibility, though not at this point a reality.

Even if you take this possibility completely seriously, though, there’s another problem with the Economist’s description: it’s not clear that this new theory would be a non-string theory!

The main people behind the amplituhedron are pretty positively disposed to string theory. If you asked them, I think they’d tell you that, rather than replacing string theory, they expect to learn more about string theory: to see how it could be reformulated in a way that yields insight about trickier problems. That’s not at all like the other “non-string theories of everything” in that list, which frame themselves as alternatives to, or even opponents of, string theory.

It is a lot like several other research programs, though, like ER=EPR and It from Qubit. Researchers in those programs try to use physical principles and toy models to say fundamental things about quantum gravity, trying to think about space and time as being made up of entangled quantum objects. By that logic, they belong in that list in the article alongside the amplituhedron. The reason they aren’t is obvious if you know where they come from: ER=EPR and It from Qubit are worked on by string theorists, including some of the most prominent ones.

The thing is, any reason to put the amplituhedron on that list is also a reason to put them. The amplituhedron is not a theory of everything, it is not at present a theory of quantum gravity. It’s a research direction that might shed new insight about quantum gravity. It doesn’t explicitly involve strings, but neither does It from Qubit most of the time. Unless you’re going to describe It from Qubit as a “non-string theory of everything”, you really shouldn’t describe the amplituhedron as one.

The amplituhedron is a really cool idea, one with great potential. It’s not something like loop quantum gravity, or causal dynamical triangulations, and it doesn’t need to be. Let it be what it is, please!

Amplitudes 2021 Retrospective

Phew!

The conference photo

Now that I’ve rested up after this year’s Amplitudes, I’ll give a few of my impressions.

Overall, I think the conference went pretty well. People seemed amused by the digital Niels Bohr, even if he looked a bit like a puppet (Lance compared him to Yoda in his final speech, which was…apt). We used Gather.town, originally just for the poster session and a “virtual reception”, but later we also encouraged people to meet up in it during breaks. That in particular was a big hit: I think people really liked the ability to just move around and chat in impromptu groups, and while nobody seemed to use the “virtual bar”, the “virtual beach” had a lively crowd. Time zones were inevitably rough, but I think we ended up with a good compromise where everyone could still see a meaningful chunk of the conference.

A few things didn’t work as well. For those planning conferences, I would strongly suggest not making a brand new gmail account to send out conference announcements: for a lot of people the emails went straight to spam. Zulip was a bust: I’m not sure if people found it more confusing than last year’s Slack or didn’t notice it due to the spam issue, but almost no-one posted in it. YouTube was complicated: the stream went down a few times and I could never figure out exactly why, it may have just been internet issues here at the Niels Bohr Institute (we did have a power outage one night and had to scramble to get internet access back the next morning). As far as I could tell YouTube wouldn’t let me re-open the previous stream so each time I had to post a new link, which probably was frustrating for those following along there.

That said, this was less of a problem than it might have been, because attendance/”viewership” as a whole was lower than expected. Zoomplitudes last year had massive numbers of people join in both on Zoom and via YouTube. We had a lot fewer: out of over 500 registered participants, we had fewer than 200 on Zoom at any one time, and at most 30 or so on YouTube. Confusion around the conference email might have played a role here, but I suspect part of the difference is simple fatigue: after over a year of this pandemic, online conferences no longer feel like an exciting new experience.

The actual content of the conference ranged pretty widely. Some people reviewed earlier work, others presented recent papers or even work-in-progress. As in recent years, a meaningful chunk of the conference focused on applications of amplitudes techniques to gravitational wave physics. This included a talk by Thibault Damour, who has by now mostly made his peace with the field after his early doubts were sorted out. He still suspected that the mismatch of scales (weak coupling on the one hand, classical scattering on the other) would cause problems in future, but after his work with Laporta and Mastrolia even he had to acknowledge that amplitudes techniques were useful.

In the past I would have put the double-copy and gravitational wave researchers under the same heading, but this year they were quite distinct. While a few of the gravitational wave talks mentioned the double-copy, most of those who brought it up were doing something quite a bit more abstract than gravitational wave physics. Indeed, several people were pushing the boundaries of what it means to double-copy. There were modified KLT kernels, different versions of color-kinematics duality, and explorations of what kinds of massive particles can and (arguably more interestingly) cannot be compatible with a double-copy framework. The sheer range of different generalizations had me briefly wondering whether the double-copy could be “too flexible to be meaningful”, whether the right definitions would let you double-copy anything out of anything. I was reassured by the points where each talk argued that certain things didn’t work: it suggests that wherever this mysterious structure comes from, its powers are limited enough to make it meaningful.

A fair number of talks dealt with what has always been our main application, collider physics. There the context shifted, but the message stayed consistent: for a “clean” enough process two or three-loop calculations can make a big difference, taking a prediction that would be completely off from experiment and bringing it into line. These are more useful the more that can be varied about the calculation: functions are more useful than numbers, for example. I was gratified to hear confirmation that a particular kind of process, where two massless particles like quarks become three massive particles like W or Z bosons, is one of these “clean enough” examples: it means someone will need to compute my “tardigrade” diagram eventually.

If collider physics is our main application, N=4 super Yang-Mills has always been our main toy model. Jaroslav Trnka gave us the details behind Nima’s exciting talk from last year, and Nima had a whole new exciting talk this year with promised connections to category theory (connections he didn’t quite reach after speaking for two and a half hours). Anastasia Volovich presented two distinct methods for predicting square-root symbol letters, while my colleague Chi Zhang showed some exciting progress with the elliptic double-box, realizing the several-year dream of representing it in a useful basis of integrals and showcasing several interesting properties. Anne Spiering came over from the integrability side to show us just how special the “planar” version of the theory really is: by increasing the number of colors of gluons, she showed that one could smoothly go between an “integrability-esque” spectrum and a “chaotic” spectrum. Finally, Lance Dixon mentioned his progress with form-factors in his talk at the end of the conference, showing off some statistics of coefficients of different functions and speculating that machine learning might be able to predict them.

On the more mathematical side, Francis Brown showed us a new way to get numbers out of graphs, one distinct but related to our usual interpretation in terms of Feynman diagrams. I’m still unsure what it will be used for, but the fact that it maps every graph to something finite probably has some interesting implications. Albrecht Klemm and Claude Duhr talked about two sides of the same story, their recent work on integrals involving Calabi-Yau manifolds. They focused on a particular nice set of integrals, and time will tell whether the methods work more broadly, but there are some exciting suggestions that at least parts will.

There’s been a resurgence of the old dream of the S-matrix community, constraining amplitudes via “general constraints” alone, and several talks dealt with those ideas. Sebastian Mizera went the other direction, and tried to test one of those “general constraints”, seeing under which circumstances he could prove that you can swap a particle going in with an antiparticle going out. Others went out to infinity, trying to understand amplitudes from the perspective of the so-called “celestial sphere” where they appear to be governed by conformal field theories of some sort. A few talks dealt with amplitudes in string theory itself: Yvonne Geyer built them out of field-theory amplitudes, while Ashoke Sen explained how to include D-instantons in them.

We also had three “special talks” in the evenings. I’ve mentioned Nima’s already. Zvi Bern gave a retrospective talk that I somewhat cheesily describe as “good for the soul”: a look to the early days of the field that reminded us of why we are who we are. Lance Dixon closed the conference with a light-hearted summary and a look to the future. That future includes next year’s Amplitudes, which after a hasty discussion during this year’s conference has now localized to Prague. Let’s hope it’s in person!